Well-posedness for the extended Schr\"odinger-Benjamin-Ono system

TL;DR

This paper establishes local well-posedness for the extended Schrödinger-Benjamin-Ono system in certain Sobolev spaces, using energy and compactness methods, and refines Strichartz estimates to lower regularity requirements.

Contribution

It proves well-posedness for a coupled nonlinear PDE system with non-symmetric nonlinearity, extending previous results by refining analytical techniques.

Findings

Well-posedness for initial data in H^{s+1/2} x H^{s} for s > 5/4.

Modified energy methods to handle non-symmetric nonlinearities.

Refined Strichartz estimates to achieve lower regularity thresholds.

Abstract

In this work we prove that the initial value problem associated to the Schr\"odinger-Benjamin-Ono type system \begin{equation*} \left\{ \begin{array}{ll} \mathrm{i}\partial_{t}u+ \partial_{x}^{2} u= uv+ \beta u|u|^{2}, \partial_{t}v-\mathcal{H}_{x}\partial_{x}^{2}v+ \rho v\partial_{x}v=\partial_{x}\left(|u|^{2}\right) u(x,0)=u_{0}(x), \quad v(x,0)=v_{0}(x), \end{array} \right. \end{equation*} with $\beta,\rho \in \mathbb{R}$ is locally well-posed for initial data $(u_{0},v_{0})\in H^{s+\frac12}(\mathbb{R})\times H^{s}(\mathbb{R})$ for $s>\frac54$. Our method of proof relies on energy methods and compactness arguments. However, due to the lack of symmetry of the nonlinearity, the usual energy has to be modified to cancel out some bad terms appearing in the estimates. Finally, in order to lower the regularity below the Sobolev threshold $s=\frac32$, we employ a refined Strichartz estimate introduced in the Benjamin-Ono setting by Koch and Tzvetkov, and further developed by Kenig and Koenig.